The notion of asymptotic dimension was introduced by Gromov in [1]} where he also
proved that \(\delta \) -hyperbolic groups have finite asymptotic dimension. Many other groups have also
been shown to have finite asymptotic dimensions [2]}, [3]}, [4]}, [5]}. The study of the asymptotic
dimension in mapping class groups started with the work of Bell-Fujiwara [2]} who modified Gromov's
argument to show that the curve complex has finite asymptotic dimension. Masur and Minsky showed that the
curve complex is Gromov hyperbolic [7]} and that the geometry of various curve complexes are closely
linked with the coarse geometry of the mapping class group [8]}. Using these facts, Bestvina,
Bromberg and Fujiwara showed that, for surfaces of finite type, \(\operatorname{Map} (\Sigma )\)  has finite asymptotic
dimension by embedding \(\operatorname{Map} (\Sigma )\)  in a finite product of trees of curve complexes [9]}.
